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This paper proves existence of optimizers of the Stein-Weiss inequalities onCarnot groups under some conditions. The adjustment of Lions' concentrationcompactness principles to Carnot groups plays an important role in our proof.Unlike known treatment to the Hardy-Littlewood-Sobolev inequality on Heisenberggroup, our arguments relate to the powers of the weight functions.

Existence of optimizers of the Stein-Weiss inequalities on Carnot groups